[r-t] Extension of TDMMs

Philip Saddleton pabs at cantab.net
Thu Aug 12 19:13:37 UTC 2004

Richard Smith <richard at ex-parrot.com> wrote at 14:52:31 on Tue, 10 Aug 
>Warkworth Surprise Minor with a mode-3 BC extension above a mode-1 EF 
>extension gives the following surprise major and royal methods.
>  &-36-6-256-36.4-2.5.6-4.5,1
>  &-38-8-258-,1
>At higher stages, however, you always get irregular lead ends (which 
>are frequently differential hunters).

Of symmetrical methods with the treble and one other bell pivoting at 
the half-lead, the proportion that have plain bob lead ends at stage n 
(even) is 1 in (n-3)(n-5)...3, giving

Minimus 1
Minor 3
Major 15
Royal 105
Max 945

I suppose that given the number of possible Minor methods and potential 
extension paths, a chance of roughly 1 in 5000 that a random choice will 
give PB lead ends at three stages means that such an example is almost 
inevitable - but is there a sufficiently large sample for four or more? 
Does the number of additional possibilities as the stage increases 
compensate for the much smaller chance of a random occurrence?

The feature that ultimately causes irregular lead ends in the above 
example is that the extending section produces plain hunting on an 
expanding number of bells: such hunting at a faster rate than the treble 
causes each bell to turn round eventually, disrupting the coursing 
order. The same problem is evident in London.


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